带拟周期强迫项的非线性波动方程的拟周期解
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摘要
在无穷维哈密顿系统的有限维不变环的存在性方面,比如非线性哈密顿偏微分方程的有限维不变环的存在性,已经取得了丰富的成果。目前主要有两种方法用于研究非线性偏微分方程的周期解和拟周期解。第一种方法是Craig-Wayne-Bourgain法,它是由Liapunov-Schmidt约化和牛顿迭代格式发展而来的。第二种方法是无穷维KAM理论,它是由经典KAM理论发展而来的。上世纪五、六十年代由三位著名数学家Kolmogorov[35],Arnold[1]和Moser[49]建立起来的经典KAM理论是哈密顿系统理论发展的里程碑,具有划时代意义,它使太阳系的稳定性得到合理的解释,并使人们能够以一种新的方法来研究哈密顿系统。建立在2n维光滑辛流形上的经典KAM理论断言,在Kolmogorov非退化性条件下,可积系统的大多数非共振不变环面在小的摄动下保持下来。上世纪九十年代经典的KAM理论被Wayne[64]和Kuksin[36]推广到无穷维哈密顿系统,所谓无穷维哈密顿系统指的就是具有无穷多个自由度的哈密顿系统。后来,Poschel[57]对其进行了重新叙述。从某种意义上说,这两种方法是互补的。CWB方法的优点是它对第二Melnikov条件的要求变弱,使得它更适用于解重法频的情况,而缺点是它不能给出拟周期解附近的动力学性态。相比CWB方法,KAM方法具有如下优点：除了证明拟周期解的存在性,它还在解的邻域内构造了标准型,并能得到解的某些动力学性质,比如解的稳定性和零Lyapunov指数；这套方法的缺点是它对第二Melnikov条件的要求太强,使得它对重法频的情况难于解决。
在本论文中,我们讨论带拟周期强迫项的非线性波动方程utt=uxx-μu-εφ(t)h(u),μ>0 (0.1)在狄利克雷边值条件u(t,0)=0=u(t,π), -∞0)的实解析拟周期函数,h为具有如下形式的非线性实解析奇函数。当Φ(t)和h满足某些条件时,经由KAM理论,我们证明了上述方程(0.1)有很多拟周期解。
本论文主要组成部分就是变换哈密顿系统,使之满足KAM定理的要求。首先要通过一个线性拟周期变量变换,把相应的哈密顿系统的线性部分变换为常系数的线性系统。然而,在这方面并没有通用的结果。因此,这篇论文的一个重要组成部分就是无穷维线性拟周期系统的可化性。实际上,有关于无穷维线性拟周期系统的可化性的研究结果很少,目前,已知的研究结果中有D.Bambusi及S.Graffi[5],Kuksin及Eliasson[31].但这两篇文章讨论的都是Schrodinger算子,由于Schrodinger方程和波动方程相应的Hilbert尺度的同构的阶不同,所以上述两个结果不能直接用于我们的问题。
文章具体安排如下：
第一章,主要介绍了无穷维KAM理论在非线性偏微分方程的有限维环面的存在性上的应用,国内外研究现状以及本文的主要结果。
第二章,首先给出了Kuksin[36]的无穷维哈密顿系统的KAM理论,这个KAM理论在§3.2证明波动方程的可化性时将用到。其次,我们给出了Poschel[57]中的关于偏微分方程的KAM定理,此定理将用来证明我们的主要结论定理3.1.1。
第三章是本论文的主要部分。不失一般性,我们取h(u)=u+u3。在§3.2中,我们讨论波动方程相应的哈密顿函数及其可化性。首先,可将波动方程(0.1)改写为如下哈密顿系统：H=H+εG4,其中且为了简便,令Xj=Wj,Zj=ω-j,我们引入坐标(…,ω-2,ω-1,ω1,ω2,….)。我们证明了存在一个实解析典则变换∑∞0,其将哈密顿函数H变换为其中且将哈密顿函数G4变换为从而知哈密顿函数H=H+εG4被∑∞0变换为H=Ho+εG4.
在§3.3中,为了应用文献[57]中的KAM定理,我们将§3.2中的哈密顿函数变换为某个四阶的部分Birkhoff标准型。即证存在一个实解析的辛变换XF1,其将哈密顿函数H=H0+εG4变换为H o XF1=Ho+εG+εG+ε2K,其中XG,XK∈A(la,s+1),这里Gij为如下给定的系数：其中ωij(ω,ε)光滑依赖于ε和ω,且存在一个绝对常数C使得,当ε足够小时,║ωij(ω,ε)║Ω*≤Cεp,又当|Im(?)|<σ0/3,ω∈Ω时,令Ζ=(Zn+1,zn+2,…)我们有|G|=O(‖z‖4a,s),|K|=O(‖z‖6a,s).引入作用量-角变量：前述标准型变换为：这里I=(I1,…,In),A=(Gij)1≤i,j≤n,B=(Gij)1≤j≤n≤i,Z=(|Zn+1|2,|Zn+2|2,…).
通过如下关系,我们引入参数向量ξ=(ξj)1≤j≤n。及新的作用量P=(Pj)1≤j≤n：Ij=εξj+ρj,ξj∈[0,1], |ρj|<ε2,1≤j≤n.则标准型变换为：从而,哈密顿函数为：其中P=εG+εG+ε2K这里G=0(|P|2)+O(|P|║Z║).
在§3.4中,我们证明主要结论定理3.1.1。为了将Poschel[57]中的KAM定理用到我们的问题中,我们引入新的参数ω。对于任意给定的ω-∈Ω,若ω∈Ω：={ω∈Ω||ω-ω≤ε},我们引入新的参数ω：ω=ω_+εω,ω∈[0,1]m.此时,哈密顿函数为H=<ω(ξ),y>+<Ω(ξ),Z>十P这里ω(ξ)=ω(?)ω其中ω=α+ε2Aξ,Ω(ξ)=β+ε2Bξ,ξ=ω(?)ξ,y=J(?),α=(μ1,…,μn),β=(μn+1,μn+2,...).对上述哈密顿函数,应用[57]中的无穷维KAM定理可知,非线性波动方程(0.1)有如下形式的解这里fj((?),ω,ε)=λj-1εPFJ*((?),ω,ε)关于变量(?)的每个分量都以2π为周期且当j∈J,(?)∈Θ(σ0/2),ω∈Ω时,我们有|f*j((?),ω,ε)|≤C。
The problem of the existence of finite-dimensional tori for infinite-dimensional hamiltonian systems, such as nonlinear PDEs, has been extensively studied in the liter-ature. So far there are two main approaches to deal with the periodic and quasi-periodic solutions of nonlinear PDEs. The first one is the Craig-Wayne-Bourgain method(CWB method for simplicity). It is a generalization of the Liapunov-Schmidt reduction and the Newtonian method. The second approach is the infinite-dimensional KAM theory which is the extension of classical KAM theory. The classical KAM theory which is constructed by three famous mathematicians Kolmogorov [35], Arnold [1] and Moser[49] in the last century is the landmark of the development of Hamiltonian systems. It made the stabil-ity of solar system got resonable explanation and brought a new method to the study of Hamiltonian systems. The classical KAM theory which constructed on 2n-dimensional smooth manifold asserts that under the Kolmogorov non-degenerate condition, the ma-jority of the non-reasont tori of integrable system are persistent under small pertubation. In the later 1990's, the celebrated KAM theory was successfully generalized to infinite dimensional setting by Wayne[64] and Kuksin[36], where the Hamiltonian systems are those with infinite many normal frequencies. Later, Poschel[57] restated the result. The two techniques are somehow complementary. We point out the advantage of CWB method is its weaker dependence on the second Melnikov conditions, so that it is more convenient to solve the case of multiple normal frequencies. The disadvantage of CWB method is that one knows nothing on the dynamics around constructed quasi-periodic solution. Comparing with Lyapunov-Schmidt reduction method, the KAM approach has its own advantages. Besides obtaining the existence of quasi-periodic solutions it allows one to construct a local normal form in a neighborhood of the obtained solu-tions, and provides more information of the dynamics, for instance on the stability of the solutions and Lyapunov exponent be 0. The disadvantage of this approach is its stonger dependence on the second Melnikov conditions, such that it is difficult to treat the case of multiple normal frequencie.
In this paper, we will are concerned with existence of quasi-periodic solutions for quasi-periodically forced nonlinear wave equation utt=uxx-μu-μφ(t)h(u),μ> 0 (0.1) on the finite x-interval [0,π] with Dirichlet boundary conditions u(t,0)= 0= u(t,π),-∞< t<∞whereεis a small positive parameter,φ(t) is a real analytic quasi-periodic function in t with frequency vectorω=(ω1,ω2...,ωm) (?) [ρ,2ρ]m for some constantρ> 0, and the nonlinearity h is a real analytic odd function of the form It is shown that under a suitable hypothesis onφ(t) and h, there are many quasi-periodic solutions for the above equation via KAM theory.
The main step is to reduce the equation to a setting where KAM theory for PDE can be applied. This needs to reduce the linear part of Hamiltonian system to constant coefficients by a linear quasi-periodic change of variables with the same basic frequen-cies as the initial system. However, we cannot guarantee in general such reducibility. A large part of the present paper will be devoted to the proof of reducibility of an infinite-dimensional linear quasi-periodic systems. In fact, the question of reducibility of infinite-dimensional linear quasi-periodic systems is also interesting itself and remains open in the general case. There are D. Bambusi and S. Graffi[5], Kuksin and Elias-son[31] in this line. However, it would seem that their results cannot be directly applied to our problems because of the difference in the orders of corresponding morphism of the Hilbert scales between Schrodinger equations and wave equations.
The paper is organized as follows:
In Chapter 1 we introduce the existence of finite-dimensional tori for nonlinear PDEs obtained in the literature and our main work in this paper.
In Chapter 2 firstly, we give the KAM theory of infinite-dimensional Hamiltonian systems in Kuksin [36] which will applied to the reducibility of wave equations in§3.2. Secondly, we give the infinite-dimensional KAM theorem for partial differential equa-tions in P6schel[57] which will applied to prove our main result Theorem3.1.1.
In Chapter 3 we discuss the main result of this paper. Without losing generality, we assume that h(u)=u+u3. In particular, in§3.2 we discuss the Hamiltonian setting and reducibility of wave equations.Firstly, let us rewrite the wave equation(0.1) as follows H=H+εG4, where and
For convenience we introduce another coordinates(…,ω_2,ω-1,ω1,w2,…)by setting zj=ωj,zj=ω_j. There is a real analytic canonical transformation∑∞0 changes Hamiltonian H into where And Hamiltonian G4 is changed into This implies the Hamiltonian H=H+εG4 is changed by the transformation∑∞0 into H=H0+εG4.
In§3.3 we transform the Hamiltonian obtained in§3.2 into some partial Birkhoff normal form of order four for using the KAM theorem in [57]. There exists a real analytic,symplectic change of coordinates XF1 in some neighborhood of the origin on the now complex Hilbert space la,s that takes the hamiltonian H=H0+εG4 into H o XF1=Ho+εG+εG+ε2K, where XG,XG,XK∈A(la,s,la,a,s), with uniquely determined coefficients whereω(ω,ε)depends smoothly onεandωand there is an absolute constant C such that‖ωij(ω,ε)‖Ω*≤Cερforεsmall enough,and we have |G|=O(‖z‖a4,s),|K|=O(‖z‖a6,s), uniformly for |Imυ|<σo/3,ω∈Ω,Z=(zn+1,zn+2,...). We introduce the action-angle variable by setting the normal form becomes with I=(I1,…,In),A=(Gij)1≤i,j≤n,B=(Gij)1 Now let us introduce the parameter veetorξ=(ξj)1 In§3.4 we prove the main result Theorem 3.1.1.In order to apply the basic KAM theorem which is attributed to Poschel [57] to our problem,we need to introduce a new parameterωbelow. For fixedω_∈Ωarbitrarily.Forω∈Ω:={ω∈Ω||ω-ω_|≤ε},we can introduce new parameterωby the followingω=ω_+εω,ω∈[0,1]m. Hence,the Hamiltonian becomes whereω(ξ)=ω(?)ωwithω=α+ε2 Aξ,Ω(ξ)=β+ε2Bξ,andξ=ω(?)ξ, y=J(?)ρ,α=(μ1,…,μn),β=(μn+1,μn+2,…)
We apply the infinite dimensional KAM theorem in [57] to the above Hamiltonian. We have that the nonlinear wave equation(0.1) possess a solution of the form where fi(υ,ω,ε)=λ(?)ερfj*(υ,ω,ε)is of period 2πin each component ofυand for j∈(?),υ∈Θ(σ0/2),ω∈Ω,we have |fj*(υ,ω,ε)|≤C.
在本论文中,我们讨论带拟周期强迫项的非线性波动方程utt=uxx-μu-εφ(t)h(u),μ>0 (0.1)在狄利克雷边值条件u(t,0)=0=u(t,π), -∞
本论文主要组成部分就是变换哈密顿系统,使之满足KAM定理的要求。首先要通过一个线性拟周期变量变换,把相应的哈密顿系统的线性部分变换为常系数的线性系统。然而,在这方面并没有通用的结果。因此,这篇论文的一个重要组成部分就是无穷维线性拟周期系统的可化性。实际上,有关于无穷维线性拟周期系统的可化性的研究结果很少,目前,已知的研究结果中有D.Bambusi及S.Graffi[5],Kuksin及Eliasson[31].但这两篇文章讨论的都是Schrodinger算子,由于Schrodinger方程和波动方程相应的Hilbert尺度的同构的阶不同,所以上述两个结果不能直接用于我们的问题。
文章具体安排如下：
第一章,主要介绍了无穷维KAM理论在非线性偏微分方程的有限维环面的存在性上的应用,国内外研究现状以及本文的主要结果。
第二章,首先给出了Kuksin[36]的无穷维哈密顿系统的KAM理论,这个KAM理论在§3.2证明波动方程的可化性时将用到。其次,我们给出了Poschel[57]中的关于偏微分方程的KAM定理,此定理将用来证明我们的主要结论定理3.1.1。
第三章是本论文的主要部分。不失一般性,我们取h(u)=u+u3。在§3.2中,我们讨论波动方程相应的哈密顿函数及其可化性。首先,可将波动方程(0.1)改写为如下哈密顿系统：H=H+εG4,其中且为了简便,令Xj=Wj,Zj=ω-j,我们引入坐标(…,ω-2,ω-1,ω1,ω2,….)。我们证明了存在一个实解析典则变换∑∞0,其将哈密顿函数H变换为其中且将哈密顿函数G4变换为从而知哈密顿函数H=H+εG4被∑∞0变换为H=Ho+εG4.
在§3.3中,为了应用文献[57]中的KAM定理,我们将§3.2中的哈密顿函数变换为某个四阶的部分Birkhoff标准型。即证存在一个实解析的辛变换XF1,其将哈密顿函数H=H0+εG4变换为H o XF1=Ho+εG+εG+ε2K,其中XG,XK∈A(la,s+1),这里Gij为如下给定的系数：其中ωij(ω,ε)光滑依赖于ε和ω,且存在一个绝对常数C使得,当ε足够小时,║ωij(ω,ε)║Ω*≤Cεp,又当|Im(?)|<σ0/3,ω∈Ω时,令Ζ=(Zn+1,zn+2,…)我们有|G|=O(‖z‖4a,s),|K|=O(‖z‖6a,s).引入作用量-角变量：前述标准型变换为：这里I=(I1,…,In),A=(Gij)1≤i,j≤n,B=(Gij)1≤j≤n≤i,Z=(|Zn+1|2,|Zn+2|2,…).
通过如下关系,我们引入参数向量ξ=(ξj)1≤j≤n。及新的作用量P=(Pj)1≤j≤n：Ij=εξj+ρj,ξj∈[0,1], |ρj|<ε2,1≤j≤n.则标准型变换为：从而,哈密顿函数为：其中P=εG+εG+ε2K这里G=0(|P|2)+O(|P|║Z║).
在§3.4中,我们证明主要结论定理3.1.1。为了将Poschel[57]中的KAM定理用到我们的问题中,我们引入新的参数ω。对于任意给定的ω-∈Ω,若ω∈Ω：={ω∈Ω||ω-ω≤ε},我们引入新的参数ω：ω=ω_+εω,ω∈[0,1]m.此时,哈密顿函数为H=<ω(ξ),y>+<Ω(ξ),Z>十P这里ω(ξ)=ω(?)ω其中ω=α+ε2Aξ,Ω(ξ)=β+ε2Bξ,ξ=ω(?)ξ,y=J(?),α=(μ1,…,μn),β=(μn+1,μn+2,...).对上述哈密顿函数,应用[57]中的无穷维KAM定理可知,非线性波动方程(0.1)有如下形式的解这里fj((?),ω,ε)=λj-1εPFJ*((?),ω,ε)关于变量(?)的每个分量都以2π为周期且当j∈J,(?)∈Θ(σ0/2),ω∈Ω时,我们有|f*j((?),ω,ε)|≤C。
The problem of the existence of finite-dimensional tori for infinite-dimensional hamiltonian systems, such as nonlinear PDEs, has been extensively studied in the liter-ature. So far there are two main approaches to deal with the periodic and quasi-periodic solutions of nonlinear PDEs. The first one is the Craig-Wayne-Bourgain method(CWB method for simplicity). It is a generalization of the Liapunov-Schmidt reduction and the Newtonian method. The second approach is the infinite-dimensional KAM theory which is the extension of classical KAM theory. The classical KAM theory which is constructed by three famous mathematicians Kolmogorov [35], Arnold [1] and Moser[49] in the last century is the landmark of the development of Hamiltonian systems. It made the stabil-ity of solar system got resonable explanation and brought a new method to the study of Hamiltonian systems. The classical KAM theory which constructed on 2n-dimensional smooth manifold asserts that under the Kolmogorov non-degenerate condition, the ma-jority of the non-reasont tori of integrable system are persistent under small pertubation. In the later 1990's, the celebrated KAM theory was successfully generalized to infinite dimensional setting by Wayne[64] and Kuksin[36], where the Hamiltonian systems are those with infinite many normal frequencies. Later, Poschel[57] restated the result. The two techniques are somehow complementary. We point out the advantage of CWB method is its weaker dependence on the second Melnikov conditions, so that it is more convenient to solve the case of multiple normal frequencies. The disadvantage of CWB method is that one knows nothing on the dynamics around constructed quasi-periodic solution. Comparing with Lyapunov-Schmidt reduction method, the KAM approach has its own advantages. Besides obtaining the existence of quasi-periodic solutions it allows one to construct a local normal form in a neighborhood of the obtained solu-tions, and provides more information of the dynamics, for instance on the stability of the solutions and Lyapunov exponent be 0. The disadvantage of this approach is its stonger dependence on the second Melnikov conditions, such that it is difficult to treat the case of multiple normal frequencie.
In this paper, we will are concerned with existence of quasi-periodic solutions for quasi-periodically forced nonlinear wave equation utt=uxx-μu-μφ(t)h(u),μ> 0 (0.1) on the finite x-interval [0,π] with Dirichlet boundary conditions u(t,0)= 0= u(t,π),-∞< t<∞whereεis a small positive parameter,φ(t) is a real analytic quasi-periodic function in t with frequency vectorω=(ω1,ω2...,ωm) (?) [ρ,2ρ]m for some constantρ> 0, and the nonlinearity h is a real analytic odd function of the form It is shown that under a suitable hypothesis onφ(t) and h, there are many quasi-periodic solutions for the above equation via KAM theory.
The main step is to reduce the equation to a setting where KAM theory for PDE can be applied. This needs to reduce the linear part of Hamiltonian system to constant coefficients by a linear quasi-periodic change of variables with the same basic frequen-cies as the initial system. However, we cannot guarantee in general such reducibility. A large part of the present paper will be devoted to the proof of reducibility of an infinite-dimensional linear quasi-periodic systems. In fact, the question of reducibility of infinite-dimensional linear quasi-periodic systems is also interesting itself and remains open in the general case. There are D. Bambusi and S. Graffi[5], Kuksin and Elias-son[31] in this line. However, it would seem that their results cannot be directly applied to our problems because of the difference in the orders of corresponding morphism of the Hilbert scales between Schrodinger equations and wave equations.
The paper is organized as follows:
In Chapter 1 we introduce the existence of finite-dimensional tori for nonlinear PDEs obtained in the literature and our main work in this paper.
In Chapter 2 firstly, we give the KAM theory of infinite-dimensional Hamiltonian systems in Kuksin [36] which will applied to the reducibility of wave equations in§3.2. Secondly, we give the infinite-dimensional KAM theorem for partial differential equa-tions in P6schel[57] which will applied to prove our main result Theorem3.1.1.
In Chapter 3 we discuss the main result of this paper. Without losing generality, we assume that h(u)=u+u3. In particular, in§3.2 we discuss the Hamiltonian setting and reducibility of wave equations.Firstly, let us rewrite the wave equation(0.1) as follows H=H+εG4, where and
For convenience we introduce another coordinates(…,ω_2,ω-1,ω1,w2,…)by setting zj=ωj,zj=ω_j. There is a real analytic canonical transformation∑∞0 changes Hamiltonian H into where And Hamiltonian G4 is changed into This implies the Hamiltonian H=H+εG4 is changed by the transformation∑∞0 into H=H0+εG4.
In§3.3 we transform the Hamiltonian obtained in§3.2 into some partial Birkhoff normal form of order four for using the KAM theorem in [57]. There exists a real analytic,symplectic change of coordinates XF1 in some neighborhood of the origin on the now complex Hilbert space la,s that takes the hamiltonian H=H0+εG4 into H o XF1=Ho+εG+εG+ε2K, where XG,XG,XK∈A(la,s,la,a,s), with uniquely determined coefficients whereω(ω,ε)depends smoothly onεandωand there is an absolute constant C such that‖ωij(ω,ε)‖Ω*≤Cερforεsmall enough,and we have |G|=O(‖z‖a4,s),|K|=O(‖z‖a6,s), uniformly for |Imυ|<σo/3,ω∈Ω,Z=(zn+1,zn+2,...). We introduce the action-angle variable by setting the normal form becomes with I=(I1,…,In),A=(Gij)1≤i,j≤n,B=(Gij)1
We apply the infinite dimensional KAM theorem in [57] to the above Hamiltonian. We have that the nonlinear wave equation(0.1) possess a solution of the form where fi(υ,ω,ε)=λ(?)ερfj*(υ,ω,ε)is of period 2πin each component ofυand for j∈(?),υ∈Θ(σ0/2),ω∈Ω,we have |fj*(υ,ω,ε)|≤C.
引文
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[22]J. Bricmont, A. Kupiainen, A. Schenkel, Renormalization group and the Melnikov prob-lesm for PDEs, Comm. Math. Phys.,221 (2001),101-140.
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[41]P.D. Lax, Developement of singularities of solutions of nonlinear hyperbolic partial dif-ferential equations, J. Math. Phys.,5(1964),611-613.
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[50]J. Moser, Recent developments in the theory of Hamiltonian systems, SIAM. Review. 28(4)(1996),459-485.
[51]S. Marmi, J.C. Yoccoz, Some Open Problems Related to Small Divisors, Lecture Notes in Math.,vol.1784, Springer, New York,2002.
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[60]P.H. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure. Appl. Math.,31(1978),31-68.
[61]P.H. Rabinowitz, Large amplitude time periodic solutions of a semi-linear wave equation, Comm. Pure. Appl. Math.,37(1984),189-206.
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[64]C.E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys.,127(1990),479-528.
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[67]X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations,230(2006),213-274.
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[12]P. Baldi, M. Berti, Forced vibations of a nonhomogeneous string, SIAM J.Math.Anal., 40(2008),382-412.
[13]M. Berti, D. Bambusi, A Birkhooff-Lewis type theorem for some Hamiltonian PDEs, SIAM J.Math.Anal.,37(2005),83-102.
[14]M. Berti, M. Procesi, Quasi-periodic solutions of completely resonant forced wave equa-tions, Comm.Part.Diff. Eq.,31(2006),959-985.
[15]M. Berti, L. Biasco, Periodic solutions of nonlinear wave equations with non-momotons forcing terms, Rend. Mat. Acc. Naz. Lincei.,16(2005),117-124.
[16]M. Berti, L. Biasco, Forced vibations of wave equations with non-momotons nonlineari-ties, Ann. Inst. Poincare.,23(2006),439-474.
[17]H. Brezis, Periodic solutions of nonlinear vibratingstrings and duality principles, Bull. AMS.,8(1983),409-426.
[18]H. Brezis, Operateurs maximaux monotones et semigroupes de constractions dans les espaces de Hilbert, North-Holland,(1973).
[19]H. Brezis, L. Nirenberg, Forced vibrations for a nonlinear wave equation, Comm. Pure Appl. Math.,31(1978),1-30.
[20]A.I. Bobenko, S.B. Kuksin, The nonlinear Klein-Gordon equation on an interval as a perturbed sin-Gordon equation, Comm. Math. Helv.,70(1995),63-112.
[21]G.D. Birkhoff, Dynamicals Systems, it Amer. Math. Soc. Colloq. Publ., vol.IX, Amer. Math. Soc., New York,(1927).
[22]J. Bricmont, A. Kupiainen, A. Schenkel, Renormalization group and the Melnikov prob-lesm for PDEs, Comm. Math. Phys.,221 (2001),101-140.
[23]Y. Choquet-Bruhat, C. De Witt-Morette, Analysis, manifolds and physics, North Hol-land, (1982).
[24]P.R. Chernoff, J.E. Marsden, Properties of infinite-dimensional Hamiltonian systems, Lecture Notes in Math,425(1974).
[25]W. Craig, C.E. Wayne, Newton's method and periodic solutions of nonlinear wave equa-tions, Comm. Pure. Appl. Math.,46(1993),1409-1498.
[26]L.Chierchia, J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys.,211(2000),497-525.
[27]J.M. Coron, Periodic solutions of a nonlinear wave equation without assumption of mono-tonicity, Math. Ann.,262(1983),273-285.
[28]R. Dieckerhoff, E. Zehnder, Boundedness of solutions via twist theorem, Ann. Sc. Norm. Super. Pisa CI. Sci., (5)14(1987),79-95.
[29]L.H. Eliasson, Perturbation of stable invariant tori for hamiltonian systems, Ann.Scuola.Norm.Su.Pisa 15,(1988),115-147.
[30]L.H. Eliasson, Reducibility and point spectrum for linear quasi-periodic skew-products, Proceedings of ICM 1998, vol.11, Doc. Math, (extra),(1998),779-787.
[31]H.L. Eliasson, S.B. Kuksin, On reducibility of Schrodinger equations with quasiperiodic in time potentials, Comm. Math. Phys.,286(2009),125-135.
[32]J. Geng, J. You, A KAM theorem for one-dimensional SchrOdinger equation with periodic boundary conditions, J. Differential Equations,209(2005),1-56.
[33]J. Geng, Y. Yi, Quasi-periodic solutions in a nonlinear SchrOdinger equation, J. Differ-ential Equations,233(2007),512-542.
[34]Lei Jiao, Yiqian Wang, The construction of quasi-periodic solutions of quasi-periodic forced Schrodinger equation, Comm. Pure Appl. Anal,8(2009),1585-1606.
[35]A.N. Kolmogorov, On the conservation of conditionally periodic motions for a small chang in Hamilton's function, Dokl. Akad. Nauk USSR,98(1954),527-530. English translation in:Lectures Notes in Physics, Springer,93,1979.
[36]S.B. Kuksin, Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Math. Vol.1556, Springer, New York,1993.
[37]S.B. Kuksin, Elements of a qualitative theory of Hamiltonian PDEs, Doc. Math. J. DMV(Extra Volume ICM) Ⅱ 819-829.
[38]S.B. Kuksin, J. Poschel, Invariant Cantor manifolds of quasiperiodic oscillations for a nonlinear Schrodinger equation, Ann. of Math.,143(1996),149-179.
[39]S.B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl.,21(1987),192-205.
[40]S.B. Kuksin, Perturbation theory of conditionally periodic solutions of infinite dimen-sional Hamiltonian systems and its application to the Korteweg-de Vries equation, Mat. Sb.,136(1988); English transl. in Math. USSR Sb.,64(1989),397-413.
[41]P.D. Lax, Developement of singularities of solutions of nonlinear hyperbolic partial dif-ferential equations, J. Math. Phys.,5(1964),611-613.
[42]G.V. Lidskil, E.I. Shul'man, Periodic solutions of the equation utt-uxx+u3=0. Funct. Anal. Appl.,22(1988),332-333.
[43]Z. Liang, J. You, Quasi-periodic solutions for 1D Schrodinger equations with higher order nonlinearity, SIAM J.Math.Anal.,36(2005),1965-1990.
[44]Z. Liang, J. You, Quasi-periodic solutions for 1D nonlinear wave equation with a general nonlinearity,http://www.ma.utexas.edu/mp-arc/c/05/05-75.pdf.
[45]V.F. Lazutkin, KAM-theory and semiclassical approximation to eigenfunctions, To ap-pear in "Springer".
[46]J.L. Lions, Quelques methodes de resolution des problemes aux limits nonlinearies, Paris: Dunod, (1969).
[47]J.L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications, Springer, (1972).
[48]J. Moser, On invariant curves of area preservion mapping of an annulus, Nachr. Akad. Wiss. Gott. Math. Phys. KI., (1962),1-20.
[49]J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169(1967),136-176.
[50]J. Moser, Recent developments in the theory of Hamiltonian systems, SIAM. Review. 28(4)(1996),459-485.
[51]S. Marmi, J.C. Yoccoz, Some Open Problems Related to Small Divisors, Lecture Notes in Math.,vol.1784, Springer, New York,2002.
[52]K.R. Meyer, G.R. Hall, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Appl. Math. Sci.,vol.90, Springer, New York,1992.
[53]J. Poschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure. Appl. Math.,35(1982),653.
[54]J. Poschel, E. Trubowitz, Inverse spectral theory, Academic Press, Boston,1987.
[55]J. Poschel, On elliptic lower dimensional tori in Hamiltonian systems, Math. Z., 202(1989),559-608.
[56]J. Poschel, Quasi-periodic solutions for nonlinear wave equations, Comm. Math. Helv., 71(1996),269-296.
[57]J. Poschel, A KAM-theorem for some nonlinear PDEs, Ann. Sc. Norm. Super. Pisa Cl. Sci. IV Ser. (15)23(1996),119-148.
[58]P.H. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math.,20(1967),145-205.
[59]P.H. Rabinowitz, Time periodic solutions of nonlinear wave equations, Manuscripta Math.,5(1971),165-194.
[60]P.H. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure. Appl. Math.,31(1978),31-68.
[61]P.H. Rabinowitz, Large amplitude time periodic solutions of a semi-linear wave equation, Comm. Pure. Appl. Math.,37(1984),189-206.
[62]M. Reed, B. Simon, Methods of modern mathematical physics, Academic Press,Vol. 1-4(1980).
[63]H. Russmann, On an inequality for trigonometric polynomials in several variables, Anal-ysis, et cetera(P.H. Rabinowitz and E. Zehnder, eds.), Academic Press, Boston, pp.545-562.
[64]C.E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys.,127(1990),479-528.
[65]X. Yuan, Lagrange stability for Duffing-type equations, J. Differential Equations, 160(2000),94-117.
[66]X. Yuan, Invariant manifold of hyperbolic-elliptic type for nonlinear wave equation, Int. J. Math. Math. Sci.,18(2003),1111-1136.
[67]X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations,230(2006),213-274.